lmomco (version 2.4.14)

lmomst3: L-moments of the 3-Parameter Student t Distribution

Description

This function estimates the first six L-moments of the 3-parameter Student t distribution given the parameters ($\xi$, $\alpha$, $\nu$) from parst3. The L-moments in terms of the parameters are $${\lambda }_1=\xi \text{,}$$ $${\lambda }_2=2^{6-4\nu }\pi \alpha {\nu }^{1/2}\mathrm{\Gamma }(2\nu -2)/[\mathrm{\Gamma }(\frac{1}{2}\nu ){]}^4\text{, and}$$ $${\tau }_4=\frac{15}{2}\frac{\mathrm{\Gamma }(\nu )}{\mathrm{\Gamma }(\frac{1}{2})\mathrm{\Gamma }(\nu -\frac{1}{2})}{\int }_0^1\frac{(1-x{)}^{\nu -3/2}[I_x(\frac{1}{2},\frac{1}{2}\nu ){]}^2}{\sqrt{x}}\mathrm{d}x-\frac{3}{2}\text{,}$$ where $I_x(\frac{1}{2},\frac{1}{2}\nu )$ is the cumulative distribution function of the Beta distribution. The distribution is symmetrical so that ${\tau }_r=0$ for odd values of $r:r\ge 3$.

The functional relation ${\tau }_4(\nu )$ was solved numerically and a polynomial approximation made. The polynomial in turn with a root-solver is used to solve $\nu ({\tau }_4)$ in parst3. The other two parameters are readily solved for when $\nu$ is available. The polynomial based on $\log {\tau }_4$ and $\log \nu$ has nine coefficients with a residual standard error (in natural logarithm units of ${\tau }_4$) of 0.0001565 for 3250 degrees of freedom and an adjusted R-squared of 1. A polynomial approximation is used to estimate the ${\tau }_6$ as a function of ${\tau }_4$; the polynomial was based on the theoLmoms estimating ${\tau }_4$ and ${\tau }_6$. The ${\tau }_6$ polynomial has nine coefficients with a residual standard error units of ${\tau }_6$ of 1.791e-06 for 3593 degrees of freedom and an adjusted R-squared of 1.

Usage

lmomst3(para, bypoly=TRUE)

Value

An R

list is returned.

lambdas

Vector of the L-moments. First element is ${\lambda }_1$, second element is ${\lambda }_2$, and so on.

ratios

Vector of the L-moment ratios. Second element is $\tau$, third element is ${\tau }_3$ and so on.

trim

Level of symmetrical trimming used in the computation, which is 0.

leftrim

Level of left-tail trimming used in the computation, which is NULL.

rightrim

Level of right-tail trimming used in the computation, which is NULL.

source

An attribute identifying the computational source of the L-moments: “lmomst3”.

Arguments

para

The parameters of the distribution.

bypoly

A logical as to whether a polynomial approximation of ${\tau }_4$ as a function of $\nu$ will be used. The default is TRUE because this polynomial is used to reverse the estimate for $\nu$ as a function of ${\tau }_4$. A polynomial of ${\tau }_6({\tau }_4)$ is always used.

Author

W.H. Asquith with A.R. Biessen

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978--146350841--8.

See Also

parst3, cdfst3, pdfst3, quast3

Examples

lmomst3(vec2par(c(1124,12.123,10), type="st3"))